II.6 The Development of the Idea of Proof, Geometry and Proof in Eighteenth-Century Mathematics, Nineteenth-Century Mathematics and the Formal Concept of Proof: Pages 136 – 137.

In the eighteenth century Euler reformulated calculus and separated it from its geometrical roots. D’Alembert associated mathematical certainty with algebra. Lagrange in his work “Mechanique Analitique” expressed the view “One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.”

Kant refers to “visualizable intuitions” or “anschauung” in German. He draws a distinction between a philosophical argument and a geometric proof. He claims the geometric proof utilizes the concrete concept of visualizable intuitions. To Kant a geometrical proof is constrained by logic but it is much more than just a purely logical analysis of the terms involved – it includes visualizable intuitions.

By the end of the nineteenth century the concept of a proof and its role in mathematics had become deeply transformed. In 1854 Riemann in Gottingen gave a talk on “On the Hypotheses Which Lie at the Foundations of Geometry”. In the 1830’s the ideas of Bolyai and Lobachevski on non-Euclidean geometry as well as related ideas of Gauss became known. In 1822 Jean Poncelet’s treatise on projective geometry opened foundational questions. Klein and Lie studied group theoretic ideas in the 1870’s. In 1882 Moritz Pasch had an influential treatise on projective geometry that handled many foundational issues. It also closed some of the gaps in Euclidean geometry. Pasch put a much greater emphasis on the pure logical structure of the proof. He still however utilized diagrams.